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Manuscript/SI/srDFT_SC-SI.tex
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@ -273,10 +273,11 @@
\begin{document} \begin{document}
\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds} \title{A density-based basis-set correction for weak and strong correlation}
\begin{abstract} \begin{abstract}
bla bla bla youpi tralala
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -306,6 +307,7 @@ Regarding the density and its gradients, these are necessary intensive quantitie
\subsection{Property of the on-top pair density} \subsection{Property of the on-top pair density}
A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as
\begin{equation} \begin{equation}
\label{eq:def_n2}
\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, \ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation} \end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
@ -371,7 +373,7 @@ As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for
The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator
\begin{equation} \begin{equation}
\label{eq:def_f} \label{eq:def_f}
f_{\wf{A+B}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation} \end{equation}
As the summations run over all orbitals in the basis set $\Bas$, the quantity $f_{\wf{}{\Bas}}(\bfr{},\bfr{})$ is orbital invariant and therefore can be expressed in terms of localized orbitals. As the summations run over all orbitals in the basis set $\Bas$, the quantity $f_{\wf{}{\Bas}}(\bfr{},\bfr{})$ is orbital invariant and therefore can be expressed in terms of localized orbitals.
In the limit of dissociated fragments, the coulomb interaction is vanishing between $A$ and $B$ and therefore any two-electron integral involving orbitals on both the system $A$ and $B$ vanishes. In the limit of dissociated fragments, the coulomb interaction is vanishing between $A$ and $B$ and therefore any two-electron integral involving orbitals on both the system $A$ and $B$ vanishes.
@ -408,11 +410,11 @@ As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $
\section{Computational considerations} \section{Computational considerations}
The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $\ntwo_{\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively. The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $\ntwo_{\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications and if one can afford a larger memory storage, one can substantially reduce the CPU time. Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
\subsection{Computation of the on-top pair density for a CASSCF wave function} \subsection{Computation of the on-top pair density for a CASSCF wave function}
Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$. Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$.
Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen. Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen.
If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $r,s,t,u$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
\begin{equation} \begin{equation}
\label{def_n2_good} \label{def_n2_good}
\ntwo_{\wf{\Bas}{}}(\br{}) = \ntwo_{\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2 \ntwo_{\wf{\Bas}{}}(\br{}) = \ntwo_{\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
@ -420,7 +422,7 @@ If the active space is referred as the set of spatial orbitals $\mathcal{A}$ whi
where where
\begin{equation} \begin{equation}
\label{def_n2_act} \label{def_n2_act}
\ntwo_{\mathcal{A}}(\br{}) = \sum_{r,s,t,u \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{t_\uparrow}\ai{u_\downarrow}}{\wf{}{\Bas}} \phi_r (\br{}) \phi_s (\br{}) \phi_t (\br{}) \phi_u (\br{}) \ntwo_{\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
\end{equation} \end{equation}
is the purely active part of the on-top pair density, is the purely active part of the on-top pair density,
\begin{equation} \begin{equation}
@ -428,12 +430,36 @@ is the purely active part of the on-top pair density,
\end{equation} \end{equation}
and and
\begin{equation} \begin{equation}
n_{\mathcal{A}}(\br{}) = \sum_{r,s\, \in \mathcal{A}} \phi_r (\br{}) \phi_s (\br{}) n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{}) \phi_u (\br{})
\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\ai{s_\downarrow} + \aic{r_\uparrow}\ai{s_\uparrow}}{\wf{}{\Bas}} \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}}
\end{equation} \end{equation}
is the purely active one-body density. is the purely active one-body density.
Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $n_{\mathcal{A}}^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
\subsection{Computation of $\murpsibas$} \subsection{Computation of $\murpsibas$}
At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function.
One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as
\begin{equation}
\label{eq:f_good}
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
\end{equation}
where
\begin{equation}
\mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
\end{equation}
and
\begin{equation}
\mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
\end{equation}
\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$.
In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong
to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$.
As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as
\begin{equation}
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}).
\end{equation}
Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$.
\bibliography{../srDFT_SC} \bibliography{../srDFT_SC}
\end{document} \end{document}

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Manuscript/srDFT_SC.tex
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@ -273,7 +273,7 @@
\begin{document} \begin{document}
\title{A density-based basis-set correction for strong correlation} \title{A density-based basis-set correction for weak and strong correlation}
\author{Emmanuel Giner} \author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr} \email{emmanuel.giner@lct.jussieu.fr}
@ -696,7 +696,7 @@ Regarding the computational cost of the present approach, it should be stressed
\hline \hline
\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ \ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\
& aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm] & aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
& aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm] & aug-cc-pVTZ & 60.1 [ ] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
\hline \hline
& & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\ & & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\
\hline \hline